In this article, we study the Ising vectors in the vertex operator algebra V+∧ associated with the Leech lattice ∧. The main result is a characterization of the Ising vectors in V+∧. We show that for any Ising vector e in V+∧, there is a sublattice E ≅ √2E8 of ∧ such that e ∈ V+E . Some properties about their corresponding τ -involutions in the moonshine vertex operator algebra V are also discussed.We show that there is no Ising vector of σ- type in V. Moreover, we compute the centralizer CAutV(z, τe) for any Ising vector e ∈ V+∧, where z is a 2B element in AutV which fixes V+∧. On the basis of this result, we also obtain an explanation for the 1A case of an observation by Glauberman-Norton (2001), which describes some mysterious relations between the centralizer of z and some 2A elements commuting z in the Monster and the Weyl groups of certain sublattices of the root lattice of type E8.
ASJC Scopus subject areas
- 数学 (全般)